09/20/2007, 06:48 AM
Here I show the composition of terms, which are needed to compute the values for continuous tetration. In effect, the matrix-method provides one relevant column, whose entries are the coefficients for the powerseries in x, where
(1) V(x)~ * Bs = V(s^x) // basic definition
(2) V(x)~ * Bs[,1] = s^x // finally, we need only the second column
(3) V(x)~ * Bs^h[,1] = {s,x}^^h // h'th (continuous) power/iterate
and then we need only the second column of Bs to determine the result.
Let's denote the base-parameter as "s". Also
\( \hspace{24} t = h(s) \) where \( \hspace{24} s=t^{1/t} \)
\( \hspace{24} u = log(t) \)
\( \hspace{24} uh = u^h \)
The last notation was made for the analysis of the symbolically computed terms, to differentiate between the "naturally" occuring powers of u, which already occur in the terms with h=1 and the powers, which are introduced by the parameter h.
Let's call the entries of Bs[,1] (the relevant second column of Bs) b_r, where r is the row-index, beginning at zero.
Then
\( \hspace{24}
\left{s,x\right}\^\^h = \sum_{r=0}^{\infty} b_r * x^r
\)
Here I look at the structure of the b's.
From the analysis of the eigensystem-structure it is best to separate the b's in a product of f- and g- terms, where the f and g occur as entries of vectors F*G as best factorizing of the eigensystem-representation. Here F contains the parameter x and t, while G contains the parameters t, u and uh.
So that
\( \hspace{24}
b_r = f_r * g_r \)
The entries f_r are simple, shown in short at the end of the code-section.
The entries g_r are the more complicated ones.
It comes out, that each g_r can be expressed as a "nested polynomial" in t,u and uh. In effect, for each g_r exists a matrix of coefficients of dimension r (the row-index), which is independent of s,t and uh. It can be rescaled to have only integer coefficients. Call them "core-matrix for term g_r", C_r
For a single term g_r, one would start with this matrix, build polynomials in uh using C_r's rows, add these polynomials weighted by powers of u to get an intermediate value of, say, g'.
This value of g'_r is then the numerator of a fraction, where the denominator has the form of the product (u-1)*(u^2-1)*(u^3-1)... with r product terms.
The evaluation of this fraction is then, say, g"_r
These g"_r are now coefficients for powers of u, also scaled by factorials to make finally the g_r terms.
It is not needed to go more into detail here; the main focus of this explanation is to introduce the role of the "core-matrices" C_r.
If these are "known", for instance stored as constants in a program, then to build the terms for F and G, and finally of b_r is just applying the powers of t,u, and uh to their coefficients
One important information is here, that all g_r can be computed in finitely many steps.
A subsequent interesting question occurs then, looking at the rate of growth of the complexity of each term. The involved powers of the parameter u grow binomially with the row-index r. I'm wondering, whether from here an argument concerning the irrationality measure (transcendence) of the final values can be derived, based on the argument, that approximation by truncated series of b depending on r, powers of u to the exponent r^2 are involved...
But I'll leave this as speculation currently.
In the appendix an edited output of the symbolical composition of the first six terms.
Gottfried
-----------------------------------------
// ==================================================================
(1) V(x)~ * Bs = V(s^x) // basic definition
(2) V(x)~ * Bs[,1] = s^x // finally, we need only the second column
(3) V(x)~ * Bs^h[,1] = {s,x}^^h // h'th (continuous) power/iterate
and then we need only the second column of Bs to determine the result.
Let's denote the base-parameter as "s". Also
\( \hspace{24} t = h(s) \) where \( \hspace{24} s=t^{1/t} \)
\( \hspace{24} u = log(t) \)
\( \hspace{24} uh = u^h \)
The last notation was made for the analysis of the symbolically computed terms, to differentiate between the "naturally" occuring powers of u, which already occur in the terms with h=1 and the powers, which are introduced by the parameter h.
Let's call the entries of Bs[,1] (the relevant second column of Bs) b_r, where r is the row-index, beginning at zero.
Then
\( \hspace{24}
\left{s,x\right}\^\^h = \sum_{r=0}^{\infty} b_r * x^r
\)
Here I look at the structure of the b's.
From the analysis of the eigensystem-structure it is best to separate the b's in a product of f- and g- terms, where the f and g occur as entries of vectors F*G as best factorizing of the eigensystem-representation. Here F contains the parameter x and t, while G contains the parameters t, u and uh.
So that
\( \hspace{24}
b_r = f_r * g_r \)
The entries f_r are simple, shown in short at the end of the code-section.
The entries g_r are the more complicated ones.
It comes out, that each g_r can be expressed as a "nested polynomial" in t,u and uh. In effect, for each g_r exists a matrix of coefficients of dimension r (the row-index), which is independent of s,t and uh. It can be rescaled to have only integer coefficients. Call them "core-matrix for term g_r", C_r
For a single term g_r, one would start with this matrix, build polynomials in uh using C_r's rows, add these polynomials weighted by powers of u to get an intermediate value of, say, g'.
This value of g'_r is then the numerator of a fraction, where the denominator has the form of the product (u-1)*(u^2-1)*(u^3-1)... with r product terms.
The evaluation of this fraction is then, say, g"_r
These g"_r are now coefficients for powers of u, also scaled by factorials to make finally the g_r terms.
It is not needed to go more into detail here; the main focus of this explanation is to introduce the role of the "core-matrices" C_r.
If these are "known", for instance stored as constants in a program, then to build the terms for F and G, and finally of b_r is just applying the powers of t,u, and uh to their coefficients
One important information is here, that all g_r can be computed in finitely many steps.
A subsequent interesting question occurs then, looking at the rate of growth of the complexity of each term. The involved powers of the parameter u grow binomially with the row-index r. I'm wondering, whether from here an argument concerning the irrationality measure (transcendence) of the final values can be derived, based on the argument, that approximation by truncated series of b depending on r, powers of u to the exponent r^2 are involved...
But I'll leave this as speculation currently.
In the appendix an edited output of the symbolical composition of the first six terms.
Gottfried
-----------------------------------------
Code:
apt = APT_Init1(u,t,s,6)
aptLhs=APT_QsLhs(apt) // gives symbolic representation for F
aptRhs=APT_QsRhsSym(apt,1) // gives symbolic representation for G
%print Mat(aptRhs) // RHS-vector G ====================================
// I fiddled with the output and rearranged to extract the Core-matrices for each term
t*dV(u)*dV(up)*dFac(-1)( // needed rescalings
[ [1 ]
[1 ]
1*V(1/u) // weighted sums of the rows of Core-matrices
[1 - 1]
/(u - 1) // denominator
u*V(1/u)
( [1 - 3 + 2]
+ [2 - 3 + 1] )
/(u-1)/(u^2-1)
u^3*V(1/u)
(( [1 - 6 + 11 - 6] // <--- from Stirling kind 1 - mirrored
+ [5 - 18 + 18 - 5]
+ [6 - 18 + 18 - 6]
+ [6 - 12 + 7 - 1]) // <--- from Stirling kind 2 - mirrored
/(u-1)/(u^2-1)/(u^3-1)
u^6*V(1/u)
( [1 - 10 + 35 - 50 + 24]
+ [9 - 60 + 130 - 105 + 26]
+ [24 - 120 + 215 - 165 + 46]
+ [40 - 180 + 275 - 180 + 45]
+ [46 - 170 + 230 - 130 + 24]
+ [36 - 120 + 145 - 75 + 14]
+ [24 - 60 + 50 - 15 + 1] )
/(u-1)/(u^2-1)/(u^3-1)/(u^4-1)
]* dV(1/uh)
// ============== LHS- vector F ===================================
aptLhs=APT_QsLhs(apt)
%1887 = [
1,
(-t + 1) /t^1,
(-t + 1)^2/t^2,
(-t + 1)^3/t^3,
(-t + 1)^4/t^4,
(-t + 1)^5/t^5,
Gottfried Helms, Kassel
